A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm

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1 International Journal of Engineering and Applied Sciences (IJEAS) ISSN: , Volume-3, Issue-9, September 2016 A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm Dr.Arkan Jassim Mohammed, Noora Ahmed Mohammed Abstract There are many technique to approximate the dimension of fractal sets. A famous technique to approximate the fractal dimension is correlation dimension. In this paper, we propose a new approach to compute the correlation dimension of fractals generated by the Escape Time Algorithm (ETA) and computing the correlation function by Grassberger-Procaccia method, was implemented using the Matlab program. A log log graph is plotted by matlab program, the result will be an approximate polynomial of degree one (straight line) that s fits the data in a least square method. The correlation dimension of fractal object is the slope of this straight line. Index Terms ETA, geometry, GPA, IFS. I. INTRODUCTION The fractal geometry is a kind of geometry different from Euclidean geometry. The main difference is in the notation of dimension. The Euclidean geometry deals with sets which exists in integer dimension while the fractal geometry deals with sets in integer dimension. Dimension is the number of degrees of freedom available for a movement in a space. In common usage, the dimension of an object is the measurements that define its shape and size. The dimension, in physic, an expression of the character of a derived quantity in relation to fundamental quantities, without regard to its numerical value, while in mathematics, the dimension measurements are used to quantify the space filling properties of set [5]. Fractal geometry is a useful way to describe and characterize complex shapes and surfaces. The idea of fractal geometry was originally derived by Mandelbrot in 1967 to describe self-similar geometric figures such as the Von Koch curve. Fractal dimension is a key quantity in fractal geometry. The value can be a non-integer and can be used as an indicator of the complexity of the curves and surfaces. For a self-similar figure, it can be decomposed into N small parts, where each part is a reduced copy of the original figure by ratio The of a self-similar figure, it can be defied as. For curves and images that are not self-similar, there exist numerous empirical methods to compute. Results from different estimators were found to differ from each other. This is partly due to the elusive definition of, i.e., the Hausdorff-Besicovitch dimension [4]. There are many kinds of dimensions. The correlation dimension is one of the fractal dimension measurement because it permits non-integer values. In (1983) Grassbergr Dr.Arkan Jassim Mohammed, Mathematics, Al-Mustansiriya/ Sciences/ Baghdad, Iraq. Noora Ahmed Mohammed, Mathematics, Al-Mustansiriya / Sciences / Baghdad, Iraq. and Procaccia [7, 8] published an algorithm (GPA) for estimating the correlation dimension that has become very popular and has been widely used since. The correlation dimension can be calculated in real time as the fractal generated by Escape Time Algorithm by using the distances between every pair of points in the attractor set of N number of points, it is so-called correlation function. It is defined as the probability that two arbitrary points on the fractal are closer together than the sides of size of the cells which cover the fractal. The paper is organized as follows: In section 2, some background material is included to assist readers less familiar with detailed to consider. Section 3 introduces the concepts needed to understand the correlation function for calculating the correlation dimension, In section 4, we presents the Escape Time Algorithm. In section 5, the proposed method to compute the correlation dimension of fractals generated by Escape Time Algorithm. Finally, the conclusions are drawn in section 6. II. THEORETICAL BACKGROUND This section presents an overview of the major concepts and results of fractals dimension and the Iterated Function System (IFS). A more detailed review of the topics in this section are as in [1,3,6]. Fractal is defined as the attractor of mutually recursive function called Iterated Function System (IFS). Such systems consist of sets of equations, which represent a rotation, translation and, scaling. Given a metric space ), we can construct a new metric space, where is the collection of nonempty compact set of, and h is the Hausdorff metric on defined by: Definition 1 [9] A transformation on a metric space ( is said to be a contraction mapping if for some Theorem 1 (contraction mapping theorem)[3]. Let be a contraction mapping on a complete metric space ( ). Then admits a unique fixed point in and moreover for any point, the sequence converges to, where denotes the n-fold composition of. Definition 2 [6]. An IFS consists of a complete metric space and a finite set of contractive transformation with contractivity factor, for. The contractivity factor for the IFS is the maximum among }. The attractor of the IFS is the unique fixed point in of the transformation defined by for any. Definition 3 [6]. A transformation on a metric space ( is called dynamical system. It is denoted by 50

2 A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm. The orbit of a point is the sequence {. Theorem 2 [6]. Let }be an IFS with attractor. The IFS is totally disconnected if and only if, for all with. Definition 4 [6]. Let { be totally disconnected IFS with attractor The transformation, defined by, for is called the associated shift transformation on. They dynamical system is called the shift dynamical system associated with IFS. III. CORRELATION DIMENSION There are many ways to define the fractal dimension, but one of the numerically simplest and most widly used is the correlation dimension of Grassberger and Porcaccia (1983) [7,8]. The real interest of the correlation dimension is determining the dimensions of the fractal attractor. Ther are others methods of measuring dimension such that hausdorff dimension, the box- counting dimension but the correlation dimension has the advantage of being straight forwardly and quickly claculated, when only a small number of points is a vailable, and is overwhelmingly concord with other calculations dimension. Definition 5: Let { be a sequence of points in. The correlation function, is defined by: {number of pairs of points with } where is the Heaviside step function (unit step function) which has a value of either 0 or 1 and may be defined as: which acts as a counter of the number of pairs of points with Euclidean separation - distance between two points on the fractal,. The multiplier is included to normalize the count by the number of pairs of points on the fractal.. Grassberger and Procaccica [7, 8] and Barnsley [6] established that for small values of the separation distance, the correlation function has been found to follow a power law such that where is the number of cells with the edge size necessary to cover all points of the fractal object and is a positive constant The '' '' symbol in this expression is used to indicate that this is not an exact equality but is a scaling relation that expected to be valid for sufficiently large and small. After taking logarithms of each side of the scaling relation and rearranging terms, so we define The correlation dimension estimated using the least squares linear regression of and versus, then the slope of linear model represent. IV. THE ESCAPE TIME ALGORITHM [6] In this section, we study fractals generated by means of Escape time algorithms (ETA). Such fractals are computed by repeatedly applying a transformation to a given initial point in the plane, the algorithm is based on the number of iterations necessary to determine whether the orbit sequence tends to infinity or not that is, an orbit diverges when its points grow, further apart without bounds. Escape time algorithm is numerical computer graphical experiment to compare the number of iterations for the orbits of different points to escape from ball of large radius, centered at the origin. A fractal can then defined as the set whose orbit does not diverge. The Escape Time Algorithm applied to any dynamical system of the form { or {. Consider a window and a region, to which orbits of points in might escape. The result will be a ''picture'' of, where in the pixel corresponding to the point is colored according to the smallest value of the positive integer, such that. Let and be the coordinates of the lower left corner and the upper right corner respectively of a closed, filled rectangle. Let an array of points in defined by:, for where is a positive integer. Define:, where is a positive number compute a finite set of points: belonging to the orbit of, for each. The total number of points computed on an orbit is at most where is a positive integer). If the set of computed points of the orbit contain no a point in when, then the pixel corresponding to will be black color, and the computation passes to the next value of. Otherwise the pixel corresponding to is assigned a color indexed by the first integer, such that, and then the computation passes to the next value of. So, the fractal generated by calculating the orbit for each point in the plane (that is pixel on screen) and checking whether it diverges. In our work, the black and white image of the fractal generating by coloring a pixel white if it is the orbit diverges i.e.,, for some, and black if it is orbit does not i.e., for all. So, we have a closed subset of constructed by the Escape Time Algorithm, defined by: such that for all } such that is black point}. The set is called the Escape Time Fractal. The Escape Time Algorithm [6]: 1. Given, s.t, define the array of point in by,, ), for any positive integer. 2. Let be a circle centered at the origin, the set is defined such that,, where is sufficiently large number. 3. Let be a complex function with the orbit of point, where. 4. Repeat,. 51

3 IF, then is colored with the color indexed by. Else it is colored Black. End IF. 5. Change all that is not black color to the white color, such that: International Journal of Engineering and Applied Sciences (IJEAS) ISSN: , Volume-3, Issue-9, September 2016 IF > THEN Graph( ), Color ( ), = numits ENDFOR { } ENDFOR { } ENDFOR { } OUTPUT: GRAPH of m Regions END. 6. Then the set of escape time point in is defined as follows, such that is black point. Output: The set is called the fractal constructed by Escape Time Algorithm. To apply the Escape Time Algorithm for IFS we find a relationship between the dynamical system and IFS which is the shift dynamical system { associated with the IFS. So, we must follow these steps: 1. Find a dynamical system{ which is an extension of a shift dynamical system associated with the IFS, and which tends to transform points off the fractal of the IFS to new points that are further away from the fractal (this is always possible if the IFS is totally disconnected ). 2. Apply the Escape Time Algorithm, with and chosen properly. V. THE CORRELATION DIMENSION OF FRACTAL GENERATED BY ESCAPE TIME ALGORITHM In this section, we propose on approach algorithm to compute correlation dimension of fractals generated by Escape Time Algorithm by finding shift dynamical system associated with IFS and the correlation function has been investigated by Grassbger Procaccia Algorithm[7,8]. A log-log graph of the correlation function versus the distances between every pair of points in the fractals constructed by Escape Time Algorithm is an approximate of the correlation dimension, i.e., the correlation dimension is deduced from the slope of the straight line scaling grid in a plot of versus. We presented and implemented in the Matlab program listed in Appendix for computing the correlation dimension of fractals generated by Escape Time Algorithm. The Proposed Algorithm (1): INPUT: READ, { = number of regions, = value } FOR PROCESS: FOR TO READ ENDFOR { } READ,,,,,, numits TO FOR TO M FOR n=1 TO nmits IF ( {,, <, >} ) and (y{,, <, >} ) THEN, ENDIF { Calculate correlation dimension (CorD) } Step1: INPUT Nt, N {Nt=No. of transients, N=NO. of points} Step2: Generate random Step3:, Step4: Step5: Step6: ] Step7: Step8: OUTPUT: PRINT (CorD),, In the following, this algorithm have been applied to the examples of fractals generated by Escape Time Algorithm. That is, we calculated the correlation dimension of fractal generated by Escape Time Algorithm. Example 1: Consider the IFS where The attractor of the IFS is the fractal Sierpinski triangle. The relationship between the dynamical system and fractals IFS is a shift dynamical system associated with IFS. Calculate the inverse transformations for the above system, which are as follows: Define, by By applying Algorithm (1) to this dynamical system with where ( and and,, and is larger enough we get the fractal set constructed by this algorithm defined by :. That is, the black point represents the set. See figure (1.a) and we find the correlation dimension by this figure (1.b) show the plot of against for the fractal. 52

(C) (C) A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm correlation dimension is computed from the sloped linear portion of the plot determined from a least square fit.

4 (C) (C) A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm correlation dimension is computed from the sloped linear portion of the plot determined from a least square fit Figure (1.a) The Sierpinski triangle In the following log-log graph indicate the correlation dimension of the Sierpinski triangle generated by a computer Matlab program Figure (2.a): The Cantor set Figure (1.b) Figure (2.b) Example 3: Consider the IFS where Example 2: Consider the IFS where The attractor of the IFS is the fractal Cantor set. The inverse transformations for the above system, which are as follows: Define, by The attractor of the IFS is the fractal. The inverse transformations for the above system, which are as follows: Define, by In the same manner as in example (1), we calculate the correlation dimension of cantor set generated by Escape Time Algorithm, ( 53

$(C) (C) In the same manner as in example (1), we calculate the correlation dimension of the fractal T generated by Escape Time Algorithm, ( 1 0.9 0.$

5 (C) (C) In the same manner as in example (1), we calculate the correlation dimension of the fractal T generated by Escape Time Algorithm, ( International Journal of Engineering and Applied Sciences (IJEAS) ISSN: , Volume-3, Issue-9, September In the same manner as in example (1), we calculate the correlation dimension of sierpinski carpet generated by Escape Time Algorithm, ( Figure (3.a) Figure (4.a): The Sierpinski Carpet Figure (3.b) Example 4: Consider the IFS where Figure (4.b) Example 5: Consider the IFS where The attractor of the IFS is the fractal Sierpinski Carpet. The inverse transformations for the above system, which are as follows: Define, by The attractor of the IFS is the fractal G. The inverse transformations for the above system, which are as follows: Define 54

6 A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm In the same manner as in example (1), we calculate the correlation dimension of fractal G generated by Escape Time Algorithm. ( (C) Figure (5.a) Figure (5.b) VI. CONCLUSIONS There are many approaches to compute the fractal dimension of an object. In this work we have presented an algorithm for computing correlation dimension of fractal generated by an Escape Time Algorithm (ETA) which is based on computing the correlation function. Computing the correlation function based on the selection of the Euclidean distance is presented and implemented the Matlab program listed in the Appendix for reducing the computational complexity of the Grassberger-procaccia Algorithm. APPENDIX Computer program for computing the correlation dimension of fractal generated by Escape Time Algorithm { Graph the Shapes using ETA } clc; clear all; %a=1/3; b=0; c=0; d=1/3; a=0; b=0; c=1; d=1; numits = 4; r = 1; M=100; A1=0; t=0; h=0; method = '1'; for p= 1:M for q=1:m x = a + (c-a)*p/m; y = b + (d-b)*q/m; n = 0; while n <= numits n = n + 1; switch lower(method) case '1' %disp('sierpinski Triangle') if (x>=0)&&(y>0.5) x=2*x; y=2*y-1; if (x>0.5)&&(y<=0.5) x=2*x-1; y=2*y; if (x<=0.5) && (y<=0.5) x=2*x; y=2*y; case '2' %disp('ex2') if (x<0.5) x1=x+y-1/2; y1=4/3*y-4/3*x+1/6; x=x1; y=y1; if (x>=0.5) x1=x-y+1/2; y1=4/3*x+4/3*y-7/6; x=x1; y=y1; case '3' %disp('ex11') if (x<=1/4) if (y<=1/4) x=4*x; y=4*y; if (y>=3/4) x=4*x; y=4*y-3; if (y>1/4)&&(y<3/4) x=2*x-1/2; y=2*y-1/2; else if (y<=1/4) x=4*x-3; y=4*y; if (y>=3/4) x=4*x-3; y=4*y-3; if (y>1/4)&&(y<3/4) x=2*x-1/2; y=2*y-1/2; case '4' %disp('cantor Set') Ex13 if (x<=0.5) x=3*x; y=3*y; else x=3*x-2; y=3*y; 55

7 case '5' %disp('ex16'); %disp('sierpinski Carpet') if (x<=1/3) if (y<=1/3) x=3*x; y=3*y; if (y>=2/3) x=3*x; y=3*y-2; if (y>1/3)&&(y<2/3) x=3*x; y=3*y-1; if x>=2/3 if (y<=1/3) x=3*x-2; y=3*y; if (y>=2/3) x=3*x-2; y=3*y-2; if (y>1/3)&&(y<2/3) x=3*x-2; y=3*y-1; if (x>=1/3)&&(x<=2/3) if (y<=2/3) x=3*x-1; y=3*y; if (y>=2/3) x=3*x-1; y=3*y-2; if (y>1/3)&&(y<2/3) x=3*x-2; y=3*y-1; otherwise disp('unknown method.') end if x^2 + y^2 > r t=t+1; P(t)=p; Q(t)=q; X(t)=x; Y(t)=y; n = numits+1; %fprintf('(%4.3f,%4.3f)=%4.3f\n',x,y,x+y); else h=h+1; P1(h)=p; Q1(h)=q; if (x^2 + y^2)^0.5 > R^2 A1 = A1 + 1; figure(1) plot(p(1:t),q(1:t),'.black','markersize',5); xlabel('\itx_n') ylabel('\ity_n') title(' Fractal generated by Escape Time Algorithm '); { Calculate correlation Dimensions } clear all;clc; International Journal of Engineering and Applied Sciences (IJEAS) ISSN: , Volume-3, Issue-9, September 2016 Ntrans=1000; N_pts=3000; x0=0; y0=0; m=3; a=[ ; ; ]; px=y0; py=x0; for j=1:ntrans s=floor(m*rand+1); nx=a(s,1)*px+a(s,2)*py+a(s,5); ny=a(s,3)*px+a(s,4)*py+a(s,6); px=nx; py=ny; nx ny x=zeros(n_pts,1); y=zeros(n_pts,1); x(1)=nx; y(1)=ny; for j=1:n_pts-1 s=floor(m*rand+1); x(j+1)=a(s,1)*x(j)+a(s,2)*y(j)+a(s,5); y(j+1)=a(s,3)*x(j)+a(s,4)*y(j)+a(s,6); x y figure(1); axis tight plot(x(1:n_pts),y(1:n_pts),'.b','markersize',3); xlabel('\itx_n') ylabel('\ity_n') title(' Fractal generated by Escape Time Algorithm'); grid ED=sparse(N_pts,N_pts); k=0; for j=1:100 %N_pts for i=j+1:100 %N_pts d=(x(i)-x(j))^2+(y(i)-y(j))^2; ED(i,j)=d; k=k+1; if mod(k,250000)==0 fprintf('%d - %3.2f\n',k,d); ED=sqrt(ED); min_eps=double(min(min(ed+(1000*ed==0)))); m_eps=double(max(max(ed))); max_eps=2^ceil(log(m_eps)/log(2)); n_div=floor(double(log(max_eps/min_eps)/log(2))); n_eps=n_div+1; eps_vec=max_eps*2.^(-((1:n_eps)'-1)); Npairs=N_pts*(N_pts-1)/2; c_eps=[]; for i=1:n_eps eps=eps_vec(i); N = (ED<eps & ED>0); S =double(sum(sum(n))); c_eps = [c_eps; S/Npairs]; omit_pts=3; k1=omit_pts+1; k2=n_eps-omit_pts; in_grid = k1:k2; xd=log(eps_vec)/log(2); 56

8 A New Approach for Finding Correlation Dimension Based on Escape Time Algorithm yd=log(c_eps)/log(2); xp=xd(in_grid); yp=yd(in_grid); [coeff,temp]=polyfit(xp,yp,1); D_Cor=coeff(1); poly=d_cor*xd+coeff(2); D_Cor figure(2); plot(xd,yd,'s-'); hold on plot(xd,poly,'r-'); axis tight plot([xd(k1),xd(k1)],[-30,30],'m--'); plot([xd(k2),xd(k2)],[-30,30],'k--'); xlabel('log_2(\epsilon)'); ylabel('log_2(c(\epsilon))'); title({'correlation dimension for fractal generated by Escape Time Algorithm'}); grid REFERENCES [1]. B. B. Mandelbrot, ''The Fractal Geometry of Nature'', W. H. Freeman and Co.,, New York, NY 480. (1983). [2]. D. Gulick, ''Encounters with Chaos'', McGraw-Hill, Inc., (1992). [3]. G. A. Edgar, ''Measure, Topology and Fractal Geometry'', New York : Springer-Verlag, (1990). [4]. G. Zhou, N.S.N. Lam, ''A comparision of fractal dimension estimations based on multiple surface generation, computers and Geosciences, 31 (2005), [5]. K. Falcone, ''Fractal Geometry, Mathematical Foundations and Applications'', John-Wiley and Sons Ltd, England (1999). [6]. M. F. Barnsley, ''Fractals. Every Where'', Academic Press Professional, Inc., San Diego, CA, USA, (1993). [7]. P. Grassberger, and I. Procaccia, ''Characterization of strange attractors, Physical. Review Letters, 50 (1983a), [8]. Grassberger, P, and Procaccia I., ''Measuring the strangeness of strange attractors. Physica D, 9 (1983b), [9]. S. N. Elaydi, ''Discrete Chaos'', Champan and Hall/CRC, (2000) 57

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